Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data

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Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data

Rosane Ushirobira, Wilfrid Perruquetti, Mamadou Mboup, Michel Fliess

To cite this version:

Rosane Ushirobira, Wilfrid Perruquetti, Mamadou Mboup, Michel Fliess. Algebraic parameter es-

timation of a multi-sinusoidal waveform signal from noisy data. European Control Conference, Jul

2013, Zurich, Switzerland. �hal-00819048�

Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data*

Rosane Ushirobira

¹

, Wilfrid Perruquetti

²

, Mamadou Mboup

³

and Michel Fliess

⁴

Abstract— In this paper, we apply an algebraic method to estimate the amplitudes, phases and frequencies of a biased and noisy sum of complex exponential sinusoidal signals. Let us stress that the obtained estimates are integrals of the noisy measured signal: these integrals act as time varying filters.

Compared to usual approaches, our algebraic method provides a more robust estimation of these parameters within a fraction of the signal’s period. We provide some computer simulations to demonstrate the efficiency of our method.

I. I

NTRODUCTION

Numerous practical engineering problems involve the es- timation of the frequencies of a biased and noisy sum of complex exponential sinusoidal signals, e.g. signal demodu- lation in communications, regulation of electronic converters power, the circadian rhythm of biological cells and the modal identification for flexible structures (see [51]). A very motivating example, developed in [42], is the position reconstruction of a human body in the sagittal plane using only accelerometer measurements.

Several different methods have been elaborated to solve this particular estimation problem, (see [25], [49] for sur- veys), such as linear regression [5], [41], adaptive least square method [47], subspace methods (high resolution) [12], [22], [46], [24], the extended Kalman filter introduced in [26], [27], [28] and refined in [2] where a simple tuning rule is given, the notches filter introduced simultaneously in [21] and [44] providing biased estimates of the frequency for standard notch (see [48]) with a first improvement obtained in [1] and an adaptive version in [9] (see also [45]), adaptive sogi-filters [13], techniques borrowed from adaptive nonlin- ear control [23], [38] or alternatively [29], [30] and more recently [3], [6], [7], [56]. Let us stress that almost all the above mentioned results (except [25], [5], [13] that needs half of the period to recover the parameters and [51] that uses also

*This work was supported by Inria Lille - Nord Europe, France. It was partially supported by the European INTERREG IV A 2 Mers Seas Zeen Cross-border Cooperation Program 2007-2013 under SYSIASS 6-20 project. It was also partially supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the Contrat de Projets Etat Region (CPER) CIA 2007-2013.

1Rosane Ushirobira is with Non-A, Inria & Institut de Math´ematiques de Bourgogne - Universit´e de Bourgogne, France (e- mail:Rosane.Ushirobira@inria.fr).

2 Wilfrid Perruquetti is with Universit´e Lille Nord de France & ´Ecole Centrale de Lille & Non-A, Inria & LAGIS (CNRS), France (e-mail:

Wilfrid.Perruquetti@inria.fr).

3 Mamadou Mboup is with CReSTIC - Universit´e de Reims Champagne Ardenne & Non-A, Inria, France (e-mail:

Mamadou.Mboup@univ-reims.fr).

4 Michel Fliess is with LIX (CNRS), ´Ecole polytechnique (e-mail:

Michel.Fliess@polytechnique.edu) & AL.I.E.N., Nancy, France, www.alien-sas.com

algebraic techniques for a single sinusoidal) deal only with the frequency estimation problem: here our method can be extended to estimate all the parameters including amplitudes and phases (see example in sections IV and V). Nevertheless, obtaining a robust estimation in the presence of noise and an unknown constant bias, continues to be an issue not quite solved.

Other interesting feature of our algebraic approach is that it is fast and online. Therefore, a comparison procedure with offline methods, such as the maximum likelihood estimation (see [11]), does not apply.

The algebraic methods used in this paper are inspired by the fundamental work of M. Fliess et al. [20], [18], [17], [19], [15], [35]. For more results in practical examples, we refer to [40], [50], [51], [53].

The parameter estimation problem for a finite sum of sinusoidal functions was notably studied by G. Riche de Prony in his 1795 seminal paper [43] (see also [24], [41]).

In this paper, we are interested in Prony’s problem for the estimation of frequencies in a sum of complex sinusoidal functions. In other words, our first goal is to estimate the frequencies of the signal

x(t) = X

n k=1

α

k

exp (i (ω

k

t + φ

k

)) (1)

from the biased and noisy output measure y(t) = x(t) + β + $

where β is an unknown constant bias and $ is a noise.

¹

Let us remark that, in its generality, the problem of param- eter estimation problem for x(t) consists on the estimation of the triplet (α

k

, ω

k

, φ

k

) for all k. For the sake of brevity, we present here only the frequencies estimation in the general case. However, an example in the case n = 3 is treated at the end of this paper, where we also indicate how to proceed for the calculation of the triplets (amplitude, frequencies, phases).

II. P

ROBLEM FORMULATION

To formulate the problem of the parametric estimation, we start with a signal depending on a set of parameters. We wish to estimate some of these parameters. They form a vector that denoted by Θ. Based on the observed noisy signal, our aim is to obtain a “good” approximation of Θ.

1Here, the noise is interpreted as afast oscillationand it does not depend on any probabilistic modeling, as in [14], [15].

Let us denote by θ

k

(1 ≤ k ≤ n) a multiple of the elementary symmetric polynomial in n variables ω

1

, . . . , ω

n

:

θ

k

:= (−i)

^k

X

1≤j1<j2<···<jk≤n

ω

j₁

ω

j₂

. . . ω

j_k

. (2)

That means that θ

1

, . . . , θ

n

can be obtained as the coeffi- cients of the polynomial in the variable X given by

Y

n

`=1

(X − iω

`

) = X

ⁿ

+ θ

1

X

ⁿ⁻¹

+ θ

2

X

ⁿ⁻²

+ · · · + θ

n

.

It is easy to see that the signal z(t) = x(t) + β and the vector Θ = {θ

1

, . . . , θ

n

} satisfy a linear differential algebraic relation provided by the differential equation:

z

⁽ⁿ⁾

(t) +

n−1

X

k=0

θ

k+1

z

^(k)

(t) − θ

n

β = 0. (3)

The Laplace transform applied on the equation (3) gives the following relation in the operational domain:

s s

ⁿ

+

n−1

X

k=0

θ

k+1

s

^k

!

Z (s) − s

²

z

⁽ⁿ⁻¹⁾

(0) (4)

−

n−2

X

j=0



s

^n−j

+

n−1

X

k=j+1

θ

k+1

s

^k−j



 z

^(j)

(0) − θ

n

β = 0.

We use the notation Θ

est

:= {θ

1

, θ

2

, . . . , θ

n

}, rather than Θ for the desired parameters. For 1 ≤ ` ≤ n, let us set θ

n+`

:=

−x

^(`−1)

(0). Since the bias θ

2n+1

:= β is not of interest, we also define Θ

_est

:= {θ

n+1

, . . . , θ

2n

, θ

2n+1

}, the set of undesired parameters. The frequencies ω

k

(1 ≤ k ≤ n) can be deduced from Θ

est

from a straightforward computation.

Now, consider the algebraic extensions C

Θest

:= C (Θ

est

) and C

Θest

:= C (Θ

_est

) and denote by C

Θest

[s] (respectively C

Θest

[s]) the polynomial ring in the variable s with coeffi- cients in C

Θest

(respectively in C

Θest

). We set

T (s) := s

ⁿ

+

n−1

X

k=0

θ

k+1

s

^k

= Y

n

`=1

(s − iω

`

) ∈ C

Θest

[s].

The relation below arises naturally from equation (4):

R (s, Z(s), Θ

est

, Θ

est

) := P (s) Z(s) + Q(s) = 0 (5) where P(s) = s T (s) ∈ C

Θest

[s] and

Q(s) = s

n−1

X

j=0



s

^n−j−1

+

n−1

X

k=j+1

θ

n−k

s

^k−j−1



 θ

n+j+1

−T (s)θ

2n+1

∈ C

^Θest

[Θ

_est

][s] (6) We start by eliminating Θ

_est

in equation (5). In other words, the polynomial Q(s) must be annihilated. Then we shall obtain a system of equations depending uniquely on Θ

est

. For that purpose, we proceed in three steps:

1) Algebraic elimination of Θ

_est

: we use the canonical form of the minimal Q-annihilator, the operator in

C

Θest

(s)

_d

ds

₂

that generates all differential operators annihilating Q.

2) Obtaining a system of equations on Θ

est

: the canonical forms of the differential operators generated by the minimal Q-annihilator provide a system of equations with good numerical properties in the time domain.

3) Resolution of the resulting system: to bring the equa- tions back to the time domain, we use the inverse Laplace transform

L

⁻¹

1

s

^m

d

^p

Z(s)

ds

^p

= (−1)

^p

(m − 1)!

Z

t 0

v

m−1,p

(τ )z(τ)dτ (7) with v

m,p

(τ) = (t − τ)

^m

τ

^p

, ∀ p, m ∈ N, m ≥ 1. To reduce the noisy influence in our estimation, we choose the integers m and p as small as possible.

The algebraic framework for our method is described in Section III, where we also define the minimal annihilators mentioned in the first point. In subsection III-A, we detail the canonical form of the annihilators and recall some well-known properties of the Weyl algebra. In Section IV, we give the frequencies estimation. Numerical simulations are given in Section V to illustrate the efficiency of our algebraic method, using a comparison with the modified Prony’s method.

III. A

NNIHILATORS VIA THE

W

EYL ALGEBRA

As we have seen in the preceding Section, our goal is to annihilate the polynomial Q, see (6). For that, we use differential operators, that is, polynomials in the variable

d

ds

with polynomial coefficients in the variable s. The polynomial Q has degree n, therefore it is clear that any differential operator of lowest degree, with respect to the variable

_ds^d

, greater than n annihilates Q, for example Π

1

=

s

_ds^d

− n

◦ · · · ◦ s

_ds^d

− 1

◦ s

_ds^d

and Π

2

=

_ds^dnn

. Some natural questions arise such as whether these annihilators are the same or if there exists a lower order

³

annihilator. The structure of the Weyl algebra C

Θ

(s)

d

ds

helps answering these questions.

Let us stress that this algebraic point of view is inspired by the work of M. Fliess et al. [17], [18], [19], [15], [35]

⁴

. The algebraic notions defined below are detailed in [10] and [37].

A. The Weyl Algebra

Definition 1: Let K be a field of characteristic zero. Let k ∈ N \ {0}. The Weyl algebra A

k

(K) is the C -algebra generated by p

1

, q

1

, . . . , p

k

, q

k

satisfying the relations

[p

i

, q

j

] = δ

ij

, [p

i

, p

j

] = [q

i

, q

j

] = 0, ∀ 1 ≤ i, j ≤ k where [·, ·] is the commutator defined by [u, v] := uv − vu, for all u, v ∈ A

k

(K). Sometimes we write simply A

k

.

2The polynomial ring in _ds^d with coefficients inCΘest[s]

3The order of an operatorΠ∈CΘ(s)h

d ds

i

is its degree as a polynomial in the variable _ds^d.

4Similar tools were used for numerical differentiation of noisy signal [36], [33] and spike detection [16].

A very useful realization of the Weyl algebra A

k

is to consider it as the algebra of polynomial differential operators on K[s

1

, . . . , s

k

] such that

p

i

= ∂

∂s

i

and q

i

= s

i×

· , ∀ 1 ≤ i ≤ k.

As a consequence, we can write

A

k

= K [q

1

, . . . , q

k

][p

1

, . . . , p

k

] = K [s

1

, . . . , s

k

] ∂

∂s

1

, . . . , ∂

∂s

k

(remark that the same notation is used for the variable s

i

and for the operator “multiplication by s

i

”).

A closely related algebra to A

k

(K) is defined as the differential operators on K[s

1

, . . . , s

k

] with coefficients in the rational functions field K(s

1

, . . . , s

k

). We denote it by B

k

(K), or B

k

for short. We can write

B

k

:= K (q

1

, . . . , q

k

)[p

1

, . . . , p

k

] = K (s

1

, . . . , s

k

) ∂

∂s

1

, . . . , ∂

∂s

k

.

In the case k = 1 for instance, we have A

1

= hp, q | pq − qp = 1i = K [s]

d ds

and B

1

= K (s) d

ds

Proposition 2: A basis for A

k

is given by q

^I

p

^J

| I, J ∈ N

^k

where q

^I

:= q

ⁱ₁¹

. . . q

_k^ik

and p

^J

:= p

^j₁¹

. . . p

^j_k^k

if I = (i

1

, . . . , i

k

) and J = (j

1

, . . . , j

k

).

Therefore an operator F ∈ A

k

can be written in a canonical form,

F = X

I,J

λ

IJ

q

^I

p

^J

with λ

IJ

∈ K.

Example 3: We need later the fallowing useful identity:

p

ⁿ

q

^m

= q

^m

p

ⁿ

+ X

n k=1

n i

m i

i!q

^m−i

p

ⁿ⁻ⁱ

An element F ∈ B

_k

can be similarly written as

F = X

I

λ

I

g

I

(s)p

^I

, where g

I

(s) ∈ K (s

1

, . . . , s

k

).

The order of an element F ∈ B

k

, F = P

I

λ

I

g

I

(s)p

^I

is defined as ord(F ) := max{[I| | g

I

(s) 6= 0}. Notice that the same definition holds for the Weyl algebra A

k

since A

k

⊂ B

_k

. Some important properties of A

_k

and B

_k

are given by the following propositions:

Proposition 4: A

_k

is a domain. Moreover, A

_k

is simple and Noetherian.

These properties are shared by B

_k

. Furthermore, A

_k

is neither a principal right domain, nor a principal left domain, while this is true for B

_k

:

Proposition 5: B

₁

admits a left division algorithm, that is, if F , G ∈ B

₁

, then there exists Q, R ∈ B

₁

such that F = QG + R and ord(R) < ord(G). Consequently, B

₁

is a principal left domain.

Lastly, it follows from the fact that

_ds^d

is a derivation:

Proposition 6 (Derivation): Given P

1

, P

2

∈ C[s], we have (Leibniz rule):

d

ⁿ

ds

ⁿ

(P

1

P

2

) = X

n k=0

n k

d

^k

P

1

ds

^k

d

^n−k

P

2

ds

^n−k

B. Annihilator

We set B := B

₁

( C ) = C (s)

d ds

.

Definition 7: Let Q ∈ C

Θest

[s]. A Q-annihilator w.r.t. B is an element of Ann

B

(Q) = {F ∈ B | F (Q) = 0}.

Since B is a left principal domain (see Proposition 5), then Ann

B

(Q) is a left principal ideal, i.e. it is generated by a unique Π

min

∈ B, up to multiplication by a polynomial in B. That means Ann

B

(Q) = B Π

min

. We call Π

min

a minimal Q-annihilator w.r.t. B. Remark that Ann

B

(Q) contains annihilators in finite integral form, i.e. operators with coefficients in C

₁

s

. We have the following lemmas:

Lemma 8: Consider Q(s) = s

ⁿ

, n ∈ N . A minimal Q- annihilator is given by

Π

n

= s d ds − n.

For m, n ∈ N , the operators Π

m

and Π

n

commute. Thus, one can use the following Lemma

Lemma 9: Let P

1

, P

2

∈ C

Θest

[s]. Let Π

i

be a P

i

- annihilator (i = 1, 2) such that Π

1

Π

2

= Π

2

Π

1

. Then Π

1

Π

2

is a (µP

1

+ ηP

2

)-annihilator for all µ, η ∈ C

Θest

. Now, recall that

Q(s) = s

n−1

X

j=0



s

^n−j−1

+

n−1

X

k=j+1

θ

n−k

s

^k−j−1



 θ

n+j+1

−T (s)θ

2n+1

So, the above Lemma provides a minimal Q-annihilator w.r.t. B:

Π

min

=

s d ds − n

◦ · · · ◦

s d ds − 1

◦

s d ds

. (8) From the identity in Example 3, it results:

Π

min

= s

ⁿ

d

ⁿ

ds

ⁿ

. (9)

Lemma 10: Let Q ∈ C

Θest

[s]. Then a minimal Q- annihilator w.r.t B

_Θest

is Π

min

= Q

_ds^d

−

^dQ_ds

.

IV. P

ARAMETER

E

STIMATION

The first step of the estimation is to determine a minimal Q-annihilator Π

min

. Then, we choose a suitable family of annihilators F = (Π

i

)

^r_i=1

in C (s)

d

ds

generated by Π

min

so that F applied to (5) provides a system of equations in Θ

est

. Finally, the frequencies are the solutions of this system in the time domain. A similar procedure can be also applied to estimate the remaining parameters (phases and amplitudes).

The order of the operators is one of the factors that must be taken in account when choosing the family F: it must be minimal to reduce noise sensitivity. Also, the use of finite- integral form annihilators is justified by (7). In addition, the choice of a well-balanced system of equations provides

“good” numerical properties.

Since the family F is generated by the minimal Q- annihilator that Π

min

, its elements are of the form:

Π = X

` i=0

f

i

(s) d

ⁱ

ds

ⁱ

!

Π

min

(s), (10)

with f

i

(s) ∈ C(s), ∀ 1 ≤ i ≤ `.

We have seen that a minimal Q-annihilator w.r.t. B is Π

min

= s

^{n d}_dsⁿn

, that applied on the relation (5) gives

Π

min

(P (s) Z(s)) = X

n j=0

P

j

(s) d

^j

Z(s) ds

^j

where for all 1 ≤ j ≤ n, P

j

(s) = (n + 1)!

(n − j)! s

^n+j+1

+ 1

(n − j)!

n−1

X

k=n−1−j

θ

k+1

(k + 1)!s

^k+1+j

Given that Π

min

annihilates Q, from the relation (5) follows an algebraic relation involving θ

1

, . . . , θ

n

:

X

n j=0

P

j

(s) d

^j

Z(s) ds

^j

= 0.

However, we need n independent equations to linearly iden- tify Θ

est

. One can show that this cannot be done in the operational domain, see for instance [55] for a similar proof in a low-dimensional case.

Therefore, the idea is to write the annihilator in (10) in a canonical form:

Π = X

` i=n

g

i

(s) d

ⁱ

ds

ⁱ

, with g

i

(s) ∈ C(s), ∀ n ≤ i ≤ `.

A suitable choice of the rational functions g

i

brings a consistent system of equations in the time domain : set

` = 2n − 1 and for 1 ≤ i ≤ 2n − 1, set g

i

(s) = 1 and g

k

(s) = 0, if k 6= i. Then, it suffices to solve the system. In the sequel, a very interesting example in the case of a sum of three sinusoidal waveform signals illustrates the usefulness of our algebraic method.

Remark 11: Let us note that in the case of a similar parameter estimation problem of a single noisy sinusoidal waveform signal, a consistent system can be found in the operational domain (see [54]).

Example 12: We apply our method in the case of a sum of three sinusoidal waveform signal, so n = 3. Since Q- annihilators are of the form (10), we choose ` = 2 to obtain a system with three equations. So the canonical form of the annihilator of order 6 is:

Π = g

0

(s) d

⁴

ds

⁴

+ g

1

(s) d

⁵

ds

⁵

+ g

2

(s) d

⁶

ds

⁶

,

where g

0

(s), g

1

(s), g

2

(s) ∈ C(s). Choosing g

0

(s) = 1, g

1

(s) = 0, g

2

(s) ; g

0

(s) = 0, g

1

(s) = 1, g

2

(s) = 0 and g

0

(s) = 0, g

1

(s) = 0, g

2

(s) = 1 gives three equations in the operational domain leading to the following system in the time domain:



 

 



1

6

J

1 1

6

J

2

J

3

−

₂₄¹

J

4

−

¹₆

J

5

−

¹₂

J

6 1

12

J

7 1

4

J

8

J

9



 

 





 θ

1

θ

2

θ

3



 = −



 J

10

J

11

J

12





where we set v

m,n

= v

m,n

(u) = (t − u)

^m

u

ⁿ

, for m, n ∈ N and J

i

= R

t

0

I

i

z(u) du, for 1 ≤ i ≤ 12, I

1

= 2v

3,4

− t v

3,3

I

2

= 14 v

2,4

− 14 tv

2,3

+ 3 t

²

v

2,2

I

3

= 14 v

1,4

− 21 t v

1,3

+ 9t

²

v

1,2

− t

³

v

1,1

I

4

= 9 v

3,5

− 5t v

3,4

I

5

= 18v

2,5

− 20t v

2,4

+ 5t

²

v

2,3

I

6

= 42 v

1,5

− 70 t v

1,4

+ 35t

²

v

1,3

− 5t

³

v

1,2

I

7

= 5 v

3,6

− 3t v

3,5

I

8

= 15 v

2,6

− 18t v

2,5

+ 5 t

²

v

2,4

I

9

= 30v

1,6

− 54t v

1,5

+ 30t

²

v

1,4

− 5t

³

v

1,3

I

10

= −16 v

1,3

+ 36 v

2,2

− 16v

3,1

+ v

0,4

+ v

4,0

I

11

= 20v

1,4

− 60 v

2,3

+ 40 v

3,2

− 5 v

4,1

− v

0,5

I

12

= −24 v

1,5

+ 15 v

4,2

+ 90 v

2,4

− 80 v

3,3

+ v

0,6

Finally, the expressions for θ

1

, θ

2

and θ

3

are:

θ

1

= 12 2 a J

2

− 2 b J

3

+ d J

10

Λ

θ

2

= 12 −2 a J

1

+ c J

3

+ e J

10

Λ

θ

3

= 4 b J

1

− 2 c J

2

+ f J

10

Λ

where a = 2J

9

J

11

+ J

6

J

12

, b = 3J

8

J

11

+ 2J

5

J

12

, c = 2J

7

J

11

+ J

4

J

12

, d = 4J

5

J

9

− 3J

6

J

8

, e = J

6

J

7

− J

4

J

9

, f = 3J

4

J

8

− 4J

5

J

7

, Λ = 2 d J

1

+ 2 e J

2

+ f J

3

In the general parameter estimation problem, we wish to estimate all parameters giving the amplitudes, the frequencies and the phases, that is the triplets (α

k

, ω

k

, φ

k

), 1 ≤ k ≤ 3 in (1). As mentioned in the Introduction, our algebraic method works very properly in this case. We give an idea of how to proceed:

•

define two new vectors Θ

est

= {θ

1

, θ

2

, θ

3

, θ

4

, θ

5

, θ

6

} and Θ

est

= {θ

7

} where θ

1

, θ

2

, θ

3

are as in (2), θ

4

= β − z(0), θ

5

= − z(0), ˙ θ

6

= −¨ z(0) and θ

7

= −β.

•

according with the new Θ

est

and Θ

est

, we obtain a new polynomial Q in the relation R (5):

Q(s) = T (s) θ

7

∈ C

Θest

[s]

and polynomials in C

Θest

[s]:

T (s) = s

³

+ θ

3

s

²

+ θ

2

s + θ

1

Q(s) = s

³

θ

4

+ s

²

(θ

5

+ θ

3

θ

4

) + s(θ

6

+ θ

2

θ

4

+ θ

3

θ

5

).

•

choose a new minimal Q-annihilator and estimate the frequencies {θ

1

, θ

2

, θ

3

} by a similar reasoning as above.

•

to linearly identify the remaining θ

i

define a minimal Q-annihilator with coefficients in C

Θest

(s), that means depending on the parameters θ

1

, θ

2

, θ

3

that we just calculated. In this case, the minimal Q-annihilator is

Π

^Θ_min^est

= T d

ds − T

⁰

that generates a family of annihilators of the form Π =

X

` i=0

g

i

(s) d

ds ◦ Π

^Θ_min^est

Once more, we apply the algebraic procedure and by a suitable choice of annihilators, we obtain a system of equations with coefficients depending on θ

1

, θ

2

, θ

3

that allows us to determine θ

4

, θ

5

and θ

6

.

For more details in an encouraging example on the po- sition reconstruction of a human body in the sagittal plane using only accelerometer measurements, we refer to [42].

V. S

IMULATIONS

The following figures show the estimation of parameters θ

1

, θ

2

and θ

3

concerning the results concerning the normal- ized mean values and variances. More precisely, the “true”

parameters are denoted by θ

1

, θ

2

and θ

3

and θ b

i,k

denotes the estimation of θ

i

obtained at the k-th run. The modified Prony’s method (PM) is used as a reference. Each point is obtained by averaging the results over 100 trials.

Dotted line curves represent exact values, while solid line curves show the estimations by our algebraic method and dashed line curves, the results by the modified Prony’s method.

Figure 1 shows the simulation results for the estimation of the parameters θ

1

, θ

2

and θ

3

versus the estimation time.

In figure 2, we plot θ b

i

=

₁₀₀¹

P

k

θ b

i,k

and

^var(b_θ2^θi) i

versus the estimation time. More simulation experiments will be presented in the final version.

VI. C

ONCLUSION

In this paper, we study the parameter estimation of a multi- sinusoidal waveform signal from noisy data. The methods used in this paper are of algebraic flavor. They allow a robust estimation, and very fast as well, within a fraction of the signal’s period.

We emphasize an essential point: the estimation obtained are based on integrals of measured signals. These particular integrals play the role of time-varying filters.

The efficiency of our algebraic method is illustrated by computer simulations.

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